Your airplane will handle just like gliders do on a bungee. That is, if you place the towhook the same way gliders have theirs. When the bungee is pulled back with adequate force, the glider almost instantly assumes a safe climb angle as soon as you release it. The same should happen with your airplane.

Do not attempt to launch with your engine running. Matter of fact, be very careful to work out a safe way to insure it not turning on at any time until the airplane is heading up and very far from the launcher. This situation is the one that makes the entire plan a bit questionable.

Launching off a bungee takes some degree of strength and coordination. Throw a powerful electric motor/prop into this and you see the problem. I hope you see the problem. If you don't see a problem, forget about the bungee. If you do, work out a safe plan.